The problems Tom Lau describes strongly suggest to me that he has not been taught how to steer a bike properly.
At test speeds most riding is below 30mph and most corners will be negotiated at 15 - 20 mph. At these low speeds riders can get by steering their motorcycle like a pushbike, but once the test is passed and they attempt to go faster they need to counter steer.
At low speeds motorcycles can be steered like a pushbike, i.e. turning the handlebars left to make the bike turn left and right to to make the bike turn right. Depending upon the size and weight and steering geometry of any motorcycle there is a critical speed, usually about 20 mph at which the rider must switch to counter steering in order to maintain proper control of the vehicle through corners.
With counter steering the rider applies gentle pressure pushing the right handlebar forwards in order to initiate a right hand turn and applies gentle pressure pushing the left handlebar forwards to initiate a left hand turn. By steering 'the wrong way', i.e. counter steering the bike will begin to turn and lean into the turn and once this happens the rider simply maintains neutral pressure on the bars until the apex of the turn is passed and then counter steers the other way to bring the bike back up to verticle in time for the next straight. If the bike is cranked over to the right while taking a right hand bend for example, it can be straighted up by applying gentle pressure pushing the left handlebar forwards.
If you have never consciously done this before, you can practice while riding at a modest speed [30 mph] in a staight line and by alternately applying gentle forward pressure, to one handlebar and then the other, so that the bike begins a gentle slalom motion.
Once you become proficient at this, you will have far greater confidence in corners and will be able to corner at speed.
As for instability at high speeds, this is often caused by tension leading the rider to grip the handlebars too tightly. Once you learn to counter steer, you will realise that your bike can be controlled at high speeds with only a gentle grip on the bars and if you find your bike beginning to feel unsteady as you increase speed, try to relax.
When a nervous rider experiences wind buffeting and grips the handlebars tightly in an attempt to steady his bike, the resultant rigidity causes the effects of the buffeting to be magnified and transmitted througout the bike. The rider then tenses even more and a vicious circle is created causing the bike to become increasingly unstable until the rider slows down.
By relaxing, the wind buffeting is not transmitted throughout the bike and the riders relaxed arms serve to absorb the buffeting, thereby causing the bike to become steady once more.
I hope this helps.